some qs! (1 Viewer)

[iambored]

Differentiate y=log2x

Change of base:
y = lnx
------
ln 2

Im fine with that
I just dont understand the next part:

dy/dx = 1/ln2 x(multiply) 1/x


I know that lnx differentiated = 1/x. and ln2 is a constant right?
But I dont get how they got the answer. Wouldnt u need to use the quotient rule or something?




If u cant understand this one click on the attachment!


Let h be pie!

Intergrate and find exact area between 2h/3 and 0 of:

Cosx cos2x

= [ sin x sin 2x] between 2h/3 and 0
= sin 2h/3 sin 4h/3 - 0
= sq.root3 /2 . sq.root 3/2
= sq.root 3 / 4


my prob is:
when u change it to the exact values, isnt 4pie/3 in the 3rd quad so instead of being a positive sq.root 3/2 it should be a negative?

 

[wogboy]

I just dont understand the next part:

dy/dx = 1/ln2 x(multiply) 1/x

I know that lnx differentiated = 1/x. and ln2 is a constant right?
But I dont get how they got the answer. Wouldnt u need to use the quotient rule or something?

Whenever you see a function of x divided by a constant, there is no need to apply the quotient rule, like in this case where the constant is ln2. The quotient rule is only required where a function of x is divided by a different function of x, and you need to differentiate that.

d(lnx/ln2)/dx = {d(lnx)/dx} / ln2 = (1/x) / ln2 = 1/x * 1/ln2

In fact there's nothing stopping you from applying the quotient rule, you can try it if you like. However, it's just a waste of time since if you do it you'll find that things will cancel out(differentiating a constant gives zero), and you'll get exactly the same answer as you did now. Try it and see what happens.

Let h be pie!

Intergrate and find exact area between 2h/3 and 0 of:

Cosx cos2x

First find out if the graph crosses the x-axis anywhere in this range of 0 < x < 2*pi/3. A quick sketch of it shows it doesn't, so you can just go along and integrate it:

Area = I{0 -> 2*pi/3} cosx - cos2x dx
= [ sin x 0.5*sin 2x] between 2*pi/3 and 0
= sin2*pi/3 - 0.5*sin(4*pi/3) - sin0 + 2*sin(2*0)
= sqrt(3)/2 - 0.5*(-sqrt(3)/2)
= sqrt(3) * (0.5 + 0.25)
= sqrt(3) * 3/4
= 3*sqrt(3)/4

See this thread if you're confused by the notation:
http://www.boredofstudies.org/community/showthread.php?s=&threadid=7127

 

[iambored]

oh, ok course! the constants just like having a 2 out the front or something!!
thanks!!

i'll look at the other 1 when i'm more awake - thanks 4 the help!

 

[TimTheTutor]

iambored - u r indeed right that sin(4pi/3)= -sqrt(3)/2. Therefore the bitmapped answer is wrong! Wogboy's solution is correct.

 

[iambored]

thanks guys!